Asymptotic expansion of the expected volume of the Wiener sausage in even dimensions

We consider the Wiener sausage for a Brownian motion up to time $t$ associated with a closed ball in even dimensional cases. We obtain the asymptotic expansion of the expected volume of the Wiener sausage for large $t$. The result says that the expansion has many $\log$ terms, which do not appear in odd dimensional cases.